Chapter 4. Radioactive Decay

Theory, worked examples and 9 previous-year questions with full solutions — IIT JAM, GATE, JEST, TIFR and CSIR NET-JRF, 2020 onwards.

Theory

4.1 Radioactivity and the modes of decay

Radioactivity is the spontaneous transformation of an unstable nucleus into a more stable configuration, with the emission of particles or electromagnetic radiation. It is a property of the nucleus alone: its rate is, to an excellent approximation, independent of temperature, pressure and chemical state, because these influences involve energies of electronvolts, a million times smaller than nuclear energies. Only electron capture and internal conversion, which involve atomic electrons, show tiny chemical effects (as in \({}^{7}\mathrm{Be}\)).

The principal modes are listed below. In \(\alpha\) decay, typical of heavy nuclei (\(Z>82\)), a \({}^4\mathrm{He}\) nucleus is emitted. In \(\beta^-\) decay a neutron becomes a proton (nuclei with too large \(N/Z\)); in \(\beta^+\) decay and electron capture (EC), usually of a K-shell electron, a proton becomes a neutron (too small \(N/Z\)). In \(\gamma\) decay an excited state, usually left by a preceding \(\alpha\) or \(\beta\) decay, emits a photon. The physics of each process is the subject of Chapter 5; here we need only the laws that govern when nuclei decay, which are the same for every mode. Of the three radiations, \(\alpha\) particles are stopped by a sheet of paper, \(\beta\) particles by a few millimetres of aluminium, while \(\gamma\) rays, being uncharged, are only attenuated exponentially by several centimetres of lead.

Decay	Change	Generic equation	Cause
\(\alpha\)	\(A\to A-4\), \(Z\to Z-2\)	\({}^{A}_{Z}\mathrm{X}\to{}^{A-4}_{Z-2}\mathrm{Y} + {}^{4}_{2}\mathrm{He}\)	nucleus too heavy
\(\beta^-\)	\(Z\to Z+1\)	\({}^{A}_{Z}\mathrm{X}\to{}^{A}_{Z+1}\mathrm{Y} + e^- + \bar\nu_e\)	\(N/Z\) too large
\(\beta^+\)	\(Z\to Z-1\)	\({}^{A}_{Z}\mathrm{X}\to{}^{A}_{Z-1}\mathrm{Y} + e^+ + \nu_e\)	\(N/Z\) too small
EC	\(Z\to Z-1\)	\({}^{A}_{Z}\mathrm{X} + e^-\to{}^{A}_{Z-1}\mathrm{Y} + \nu_e\)	\(N/Z\) too small
\(\gamma\)	none	\({}^{A}_{Z}\mathrm{X}^*\to{}^{A}_{Z}\mathrm{X} + \gamma\)	excess excitation energy

4.2 The law of radioactive decay

(a) Decay constant and exponential law

The decay of an individual nucleus is a random quantum event, and a nucleus does not age: the probability that it decays in a short interval \(dt\) is \(\lambda\,dt\), where the decay constant \(\lambda\) is characteristic of the nuclide and its state. For \(N\) nuclei the expected number of decays in \(dt\) is \(N\lambda\,dt\), so \[\boxed{\,\frac{dN}{dt} = -\lambda N, \qquad N(t) = N_0\,e^{-\lambda t}\,}\] A plot of \(\ln N\) against \(t\) is a straight line of slope \(-\lambda\), and the number decayed by time \(t\) is \(N_0(1 - e^{-\lambda t})\).

(b) Half-life and mean life

The half-life is the time in which \(N\) falls to half its value. Setting \(N = N_0/2\), \[\boxed{\,T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}, \qquad N(t) = N_0\,2^{-t/T_{1/2}}\,}\] so after \(n\) half-lives the fraction remaining is \((1/2)^n\), whether or not \(n\) is an integer; for instance, one third of a half-life leaves \(2^{-1/3} = 0.79\) of the nuclei. The mean life is the average lifetime. Of \(N_0\) nuclei, \(\lambda N_0e^{-\lambda t}dt\) decay between \(t\) and \(t+dt\), each having lived for \(t\), so \[\tau = \lambda\int_0^\infty t\,e^{-\lambda t}\,dt = \frac1\lambda, \qquad \boxed{\,\tau = \frac{1}{\lambda}, \qquad T_{1/2} = 0.693\,\tau\,}\] In a mean life the number falls by the factor \(e\), \(N = N_0e^{-t/\tau}\). A “lifetime” quoted without qualification is the mean life.

Exam tip. Check whether a problem gives \(T_{1/2}\) or \(\tau\). Use \(2^{-t/T_{1/2}}\) when \(t\) is a simple multiple of \(T_{1/2}\), and \(e^{-t/\tau}\) when it is a multiple of \(\tau\). When only a ratio of activities \(\mathcal{A}_0/\mathcal{A}\) at a time \(t\) is known, \(\lambda = t^{-1}\ln(\mathcal{A}_0/\mathcal{A})\).

Example 4.1 Two nuclides of half-lives 10 min and 20 min are initially equal in number. Find the ratio of the numbers decayed after 60 min.

Solution. They pass through 6 and 3 half-lives, so the fractions decayed are \(1 - 2^{-6} = 63/64\) and \(1 - 2^{-3} = 7/8\), in the ratio \(9/8\).

4.3 Activity and its units

The activity is the number of decays per unit time, \[\boxed{\,\mathcal{A} = -\frac{dN}{dt} = \lambda N = \frac{N}{\tau} = \mathcal{A}_0\,e^{-\lambda t}\,}\] It is what a detector measures, and \(\mathcal{A} = \lambda N\) is the bridge between the measured rate and the number of atoms. The SI unit is the becquerel (1 Bq = 1 decay per second); the older curie, originally the activity of 1 g of \({}^{226}\mathrm{Ra}\), is \[\boxed{\,1\ \text{Ci} = 3.7\times10^{10}\ \text{Bq}\,}\] A mass \(m\) of a nuclide of molar mass \(M\) contains \(N = mN_A/M\) atoms (\(N_A = 6.022\times10^{23}\) mol\(^{-1}\)), so \[\boxed{\,\mathcal{A} = \frac{\ln2}{T_{1/2}}\,\frac{m\,N_A}{M}\,}\] The specific activity (per unit mass) is inversely proportional to the half-life and to the molar mass: a gram of \({}^{238}\mathrm{U}\) (\(T_{1/2} = 4.47\times10^9\) y) has only \(1.2\times10^4\) Bq, while a gram of \({}^{226}\mathrm{Ra}\) (1600 y) has about 1 Ci. Convert half-lives to seconds (\(1\ \text{yr} = 3.156\times10^7\) s, \(1\ \text{day} = 86\,400\) s) before computing activities in Bq. Two natural sources are worth remembering: living carbon, with \({}^{14}\mathrm{C}/{}^{12}\mathrm{C}\approx1.3\times10^{-12}\), has about 0.25 Bq per gram of carbon, and the \({}^{40}\mathrm{K}\) in a human body (about 20 mg of \({}^{40}\mathrm{K}\) in 70 kg, \(T_{1/2} = 1.25\times10^9\) y) gives some 4–6 kBq.

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Chapter 4 Radioactive Decay